48 1. ALGEBRAIC THEMES section are analogues of results of Section 1.6 , with the proofs also follow-ing a nearly identical course. In this discussion, the degree of a polynomial plays the role that absolute value plays for integers. Proposition 1.8.6. Let f , g , h , u , and v denote polynomials in KŒxŁ . (a) If uv D 1 , then u;v 2 K . (b) If f j g and g j f , then there is a k 2 K such that g D kf . (c) Divisibility is transitive: If f j g and g j h , then f j h . (d) If f j g and f j h , then for all polynomials s;t , f j .sg C th/ . Proof. For part (a), if uv D 1 , then both of u;v must be nonzero. If either of u or v had positive degree, then uv would also have positive degree. Hence both u and v must be elements of K . For part (b), if g D vf and f D ug , then g D uvg , or g.1 ± uv/ D0 . If g D0 , then k D0 meets the requirement. Otherwise, 1 ± uv D0 , so both u and v are elements of K , by part (a), and k D v satisﬁes the requirement. The remaining parts are left to the reader.
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